4.NF 4b: How Much Discussion is Too Much Discussion? (#265)


A Year in the Life: Ambient Math Wins the Race to the Top!
Day 265

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4.  Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.  For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5.  (In general n x (a/b) = (n x a)/b.)

Isn’t 4b the same as 4a?  It seems the addition of parenthesis is the only difference, since 4b’s example: 3 x 2/5 = 6/5 = 1 1/5, is the same as 4a’s example: 5 x 1/4 = 5/4 = 1 1/4.  The addition of parenthesis introduces an algebraic concept that may be best reserved until later, therefore teaching only one concept at a time.  Math By Hand slowly and carefully introduces the basics of beginning algebra at the end of Grade 4, with stories and hands-on tools.

Having students participate in group discussion is a staple of the Common Core classroom, in both math and language arts.  I conducted the usual lesson plans search for this standard and found many, many relevant resources and websites, with most of them advocating student group learning and discussion.

At first glance, the goal of student-centered learning is a valid one.  But on closer inspection, questions arise.  Such as, is too much time devoted to repetitious discussion?  Or, is it a stretch to expect children under the age of reason (12-14 years old) to engage in logical discussion and debate?

Here is a list of questions focused on suggestions for student feedback, from the website CPALMS

Feedback to Students
1. Monitor students during individual work time by circulating and seeing what is being recorded as they work the problem. Encourage students working out a correct solution strategy and challenge them to find another way to confirm their solution. Prompt students who are struggling or not addressing the task with the following possible questions:

What do we know about the situation?

Can a drawing help you see what is happening?

Is there another way to prove your answer is correct?

What does that amount tell you?

Does this look like any problems we have done before? Why or why not?

How did you determine that amount?

What do the numbers used in the problem represent?

What does the 4 tell us? The 2/3?

How is the number of ribbons related to 2/3 yards?

What operations might we use to find a solution?

How did you decide in this task that you needed to use…?

Could we have used another operation or property to solve this task? Why or why not?

2. During the group sharing and discussion encourage students to share and listen to others. Acknowledge those asking questions of each other that promote deeper understanding. (This addresses the Math Practice Standard: Construct viable arguments and critique the reasoning of others.)

3. In preparation for the instruction after this lesson, sort the summative assessments according to understanding. Form small groups from this work. Once small groups are assembled according to understanding, point out misconceptions and use fraction bars, the area model, or number line to illustrate why the solution was incorrect.

Summative Assessment
1.At the conclusion of this lesson, students will be given an incorrect solution and asked to determine whether or not it is correct and to justify their thinking. 

I am especially troubled by the last sentence.  Why give a student an incorrect solution?  It sounds like trickery, and to what end?  Imposing rigor and deep thinking in this manner is a mistaken notion.  It’s like watching a pot boil.  Students need facts, they need to be given tools for lifelong learning, then be allowed to go to it.  Rather than having them toil endlessly over small, detailed tasks, shouldn’t we be giving them a true renaissance education, one that’s broad and motivating?

This sort of endless questioning promotes doubt and insecurity.  In the Waldorf classroom, the teacher stands before the students as a loved authority figure, one who represents the world to those in his or her care as a living font of wisdom and knowledge.  This sort of teaching is not pedantic or overbearing.  It draws forth innate knowledge from the student, but (and this is most important) it does so indirectly.  Student and teacher together bear witness to the wonder of the world, with every concept or idea taught as an element of that wonder.

In this light, fractions are wonder-ful.  They shine with their own glory and are downright, interesting fun.  When this is seen, when student interest is kindled, learning occurs naturally.  The content of the above standard is simple.  It can be taught briefly, economically, and hands-on.  With modeling, but without the repetition and inherent distrust that’s endemic in Common Core style probing and questioning.

Caroline Myss, medical intuitive, teacher and author, once relayed a story about intuitive knowing.  Years ago as a novice gardener, she found herself continually pulling carrots out of the ground to see how they were doing and track their progress.  Excellent analogy.  (See below for recipegirl.com’s wonderful portrait of multicolored carrots.)

As parents and teachers, we must let the learning process happen in a nurturing environment that is not intrusively counterproductive.  Because knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.



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